Theorem sbcbi2 3049

Description: Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)

Assertion

Ref	Expression
sbcbi2	|- ( A. x ( ph <-> ps ) -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )

Proof of Theorem sbcbi2

Step	Hyp	Ref	Expression
1		abbi 2319	. . 3 |- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } )
2		eleq2 2269	. . 3 |- ( { x | ph } = { x | ps } -> ( A e. { x | ph } <-> A e. { x | ps } ) )
3	1, 2	sylbi 121	. 2 |- ( A. x ( ph <-> ps ) -> ( A e. { x | ph } <-> A e. { x | ps } ) )
4		df-sbc 2999	. 2 |- ( [. A / x ]. ph <-> A e. { x | ph } )
5		df-sbc 2999	. 2 |- ( [. A / x ]. ps <-> A e. { x | ps } )
6	3, 4, 5	3bitr4g 223	1 |- ( A. x ( ph <-> ps ) -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371   = wceq 1373   e. wcel 2176   {cab 2191   [.wsbc 2998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-sbc 2999

This theorem is referenced by:  csbeq2  3117